Nontrivial attractor-repellor maps of $S^2$ and rotation numbers

We consider an orientation preserving homeomorphism $h$ of $S^2$ which admits a repellor denoted $\mathrm{\infty }$ and an attractor $-\mathrm{\infty }$, which is not a North-South map, such that the basins of $\mathrm{\infty }$ and $-\mathrm{\infty }$ intersect. We study various aspects of the rotation number of $h:S^2\setminus{\pm\infty}\to S^2\setminus{\pm\infty}$, especially its relationship with the existence of periodic orbits.